Two casks of 48 litres and 42 litres are filled with mixtures of milk and water, the proportions in the two casks being respectively, 13 : 7 and 18 : 17. If the contents of the two casks be mixed and 20 litres of water be added to the whole, what will be the proportion of milk and water in the resulting mixture?

Quantity of milk in the mixture,

= $$\frac{13}{20} \times 48 + \frac{18}{35} \times 42$$

= $$\frac{156}{5} + \frac{108}{5} = \frac{264}{5}$$L

Quantity of water in the mixture,

= $$\frac{7}{20} \times 48 + \frac{17}{35} \times 42$$

= $$\frac{84}{5} + \frac{102}{5} = \frac{186}{5}$$L

20L of mixture is added to the mixture,

= $$\frac{186}{5} + 20 = \frac{186 + 100}{5}$$

= $$\frac{286}{5}$$L

Required ratio = $$\frac{264}{5} : \frac{286}{5}$$

= 12 : 13

Hence, option C is the correct answer.